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148-508: In the general theory of relativity , the Einstein field equations ( EFE ; also known as Einstein's equations ) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Albert Einstein in 1915 in the form of a tensor equation which related the local spacetime curvature (expressed by the Einstein tensor ) with the local energy, momentum and stress within that spacetime (expressed by

296-400: A = 1 n ∂ Γ i j a ∂ x a − ∑ a = 1 n ∂ Γ a i a ∂ x j + ∑ a = 1 n ∑ b = 1 n ( Γ

444-478: A b ∂ x ~ a ∂ x j ∂ x ~ b ∂ x k , {\displaystyle R_{jk}={\widetilde {R}}_{ab}{\frac {\partial {\widetilde {x}}^{a}}{\partial x^{j}}}{\frac {\partial {\widetilde {x}}^{b}}{\partial x^{k}}},} so that R i j {\displaystyle R_{ij}} define

592-1338: A b a Γ i j b − Γ i b a Γ a j b ) {\displaystyle {\begin{aligned}\Gamma _{ab}^{c}&:={\frac {1}{2}}\sum _{d=1}^{n}\left({\frac {\partial g_{bd}}{\partial x^{a}}}+{\frac {\partial g_{ad}}{\partial x^{b}}}-{\frac {\partial g_{ab}}{\partial x^{d}}}\right)g^{cd}\\R_{ij}&:=\sum _{a=1}^{n}{\frac {\partial \Gamma _{ij}^{a}}{\partial x^{a}}}-\sum _{a=1}^{n}{\frac {\partial \Gamma _{ai}^{a}}{\partial x^{j}}}+\sum _{a=1}^{n}\sum _{b=1}^{n}\left(\Gamma _{ab}^{a}\Gamma _{ij}^{b}-\Gamma _{ib}^{a}\Gamma _{aj}^{b}\right)\end{aligned}}} as maps φ : U → R {\displaystyle \varphi :U\rightarrow \mathbb {R} } . Now let ( U , φ ) {\displaystyle \left(U,\varphi \right)} and ( V , ψ ) {\displaystyle \left(V,\psi \right)} be two smooth charts with U ∩ V ≠ ∅ {\displaystyle U\cap V\neq \emptyset } . Let R i j : φ ( U ) → R {\displaystyle R_{ij}:\varphi (U)\rightarrow \mathbb {R} } be

740-767: A c b . {\displaystyle \mathrm {Ric} _{ab}=\mathrm {R} ^{c}{}_{bca}=\mathrm {R} ^{c}{}_{acb}.} Sign conventions. Note that some sources define R ( X , Y ) Z {\displaystyle R(X,Y)Z} to be what would here be called − R ( X , Y ) Z ; {\displaystyle -R(X,Y)Z;} they would then define Ric p {\displaystyle \operatorname {Ric} _{p}} as − tr ⁡ ( X ↦ R p ⁡ ( X , Y ) Z ) . {\displaystyle -\operatorname {tr} (X\mapsto \operatorname {R} _{p}(X,Y)Z).} Although sign conventions differ about

888-543: A pair of black holes merging . The simplest type of such a wave can be visualized by its action on a ring of freely floating particles. A sine wave propagating through such a ring towards the reader distorts the ring in a characteristic, rhythmic fashion (animated image to the right). Since Einstein's equations are non-linear , arbitrarily strong gravitational waves do not obey linear superposition , making their description difficult. However, linear approximations of gravitational waves are sufficiently accurate to describe

1036-423: A vacuum state with an energy density ρ vac and isotropic pressure p vac that are fixed constants and given by ρ v a c = − p v a c = Λ κ , {\displaystyle \rho _{\mathrm {vac} }=-p_{\mathrm {vac} }={\frac {\Lambda }{\kappa }},} where it is assumed that Λ has SI unit m and κ

1184-428: A (0,2)-tensor field on M {\displaystyle M} . In particular, if X {\displaystyle X} and Y {\displaystyle Y} are vector fields on M {\displaystyle M} , then relative to any smooth coordinates one has The final line includes the demonstration that the bilinear map Ric is well-defined, which is much easier to write out with

1332-1073: A bilinear map Ric p : T p M × T p M → R {\displaystyle \operatorname {Ric} _{p}:T_{p}M\times T_{p}M\rightarrow \mathbb {R} } by ( X , Y ) ∈ T p M × T p M ↦ Ric p ⁡ ( X , Y ) = ∑ i , j = 1 n R i j ( φ ( x ) ) X i ( p ) Y j ( p ) , {\displaystyle (X,Y)\in T_{p}M\times T_{p}M\mapsto \operatorname {Ric} _{p}(X,Y)=\sum _{i,j=1}^{n}R_{ij}(\varphi (x))X^{i}(p)Y^{j}(p),} where X 1 , … , X n {\displaystyle X^{1},\ldots ,X^{n}} and Y 1 , … , Y n {\displaystyle Y^{1},\ldots ,Y^{n}} are

1480-570: A body in accordance with Newton's second law of motion , which states that the net force acting on a body is equal to that body's (inertial) mass multiplied by its acceleration . The preferred inertial motions are related to the geometry of space and time: in the standard reference frames of classical mechanics, objects in free motion move along straight lines at constant speed. In modern parlance, their paths are geodesics , straight world lines in curved spacetime . Conversely, one might expect that inertial motions, once identified by observing

1628-858: A calculation with the chain rule and the product rule that R i j ( x ) = ∑ k , l = 1 n r k l ( ψ ∘ φ − 1 ( x ) ) D i | x ( ψ ∘ φ − 1 ) k D j | x ( ψ ∘ φ − 1 ) l . {\displaystyle R_{ij}(x)=\sum _{k,l=1}^{n}r_{kl}\left(\psi \circ \varphi ^{-1}(x)\right)D_{i}{\Big |}_{x}\left(\psi \circ \varphi ^{-1}\right)^{k}D_{j}{\Big |}_{x}\left(\psi \circ \varphi ^{-1}\right)^{l}.} where D i {\displaystyle D_{i}}

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1776-560: A computer, or by considering small perturbations of exact solutions. In the field of numerical relativity , powerful computers are employed to simulate the geometry of spacetime and to solve Einstein's equations for interesting situations such as two colliding black holes. In principle, such methods may be applied to any system, given sufficient computer resources, and may address fundamental questions such as naked singularities . Approximate solutions may also be found by perturbation theories such as linearized gravity and its generalization,

1924-476: A conical region in the manifold will instead have larger volume than it would in Euclidean space. The Ricci curvature is essentially an average of curvatures in the planes including ξ {\displaystyle \xi } . Thus if a cone emitted with an initially circular (or spherical) cross-section becomes distorted into an ellipse ( ellipsoid ), it is possible for the volume distortion to vanish if

2072-503: A curiosity among physical theories. It was clearly superior to Newtonian gravity , being consistent with special relativity and accounting for several effects unexplained by the Newtonian theory. Einstein showed in 1915 how his theory explained the anomalous perihelion advance of the planet Mercury without any arbitrary parameters (" fudge factors "), and in 1919 an expedition led by Eddington confirmed general relativity's prediction for

2220-519: A curved generalization of Minkowski space. The metric tensor that defines the geometry—in particular, how lengths and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi- or pseudo-Riemannian metric. Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, the Levi-Civita connection , and this is, in fact,

2368-539: A curved geometry of spacetime in general relativity; there is no gravitational force deflecting objects from their natural, straight paths. Instead, gravity corresponds to changes in the properties of space and time, which in turn changes the straightest-possible paths that objects will naturally follow. The curvature is, in turn, caused by the energy–momentum of matter. Paraphrasing the relativist John Archibald Wheeler , spacetime tells matter how to move; matter tells spacetime how to curve. While general relativity replaces

2516-542: A gravitational field (cf. below ). The actual measurements show that free-falling frames are the ones in which light propagates as it does in special relativity. The generalization of this statement, namely that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames, is known as the Einstein equivalence principle , a crucial guiding principle for generalizing special-relativistic physics to include gravity. The same experimental data shows that time as measured by clocks in

2664-466: A gravitational field— proper time , to give the technical term—does not follow the rules of special relativity. In the language of spacetime geometry, it is not measured by the Minkowski metric . As in the Newtonian case, this is suggestive of a more general geometry. At small scales, all reference frames that are in free fall are equivalent, and approximately Minkowskian. Consequently, we are now dealing with

2812-450: A massive central body M is given by A conservative total force can then be obtained as its negative gradient where L is the angular momentum . The first term represents the force of Newtonian gravity , which is described by the inverse-square law. The second term represents the centrifugal force in the circular motion. The third term represents the relativistic effect. There are alternatives to general relativity built upon

2960-768: A number of exact solutions are known, although only a few have direct physical applications. The best-known exact solutions, and also those most interesting from a physics point of view, are the Schwarzschild solution , the Reissner–Nordström solution and the Kerr metric , each corresponding to a certain type of black hole in an otherwise empty universe, and the Friedmann–Lemaître–Robertson–Walker and de Sitter universes , each describing an expanding cosmos. Exact solutions of great theoretical interest include

3108-453: A problem, however, as there is a lack of a self-consistent theory of quantum gravity . It is not yet known how gravity can be unified with the three non-gravitational forces: strong , weak and electromagnetic . Einstein's theory has astrophysical implications, including the prediction of black holes —regions of space in which space and time are distorted in such a way that nothing, not even light , can escape from them. Black holes are

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3256-697: A relativistic theory of gravity. After numerous detours and false starts, his work culminated in the presentation to the Prussian Academy of Science in November 1915 of what are now known as the Einstein field equations, which form the core of Einstein's general theory of relativity. These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present. A version of non-Euclidean geometry , called Riemannian geometry , enabled Einstein to develop general relativity by providing

3404-516: A shape is deformed as one moves along geodesics in the space. In general relativity , which involves the pseudo-Riemannian setting, this is reflected by the presence of the Ricci tensor in the Raychaudhuri equation . Partly for this reason, the Einstein field equations propose that spacetime can be described by a pseudo-Riemannian metric, with a strikingly simple relationship between the Ricci tensor and

3552-416: A small cone about ξ {\displaystyle \xi } , will have smaller volume than the corresponding conical region in Euclidean space, at least provided that ε {\displaystyle \varepsilon } is sufficiently small. Similarly, if the Ricci curvature is negative in the direction of a given vector ξ {\displaystyle \xi } , such

3700-534: A specified distribution of matter and energy in the form of a stress–energy tensor, the EFE are understood to be equations for the metric tensor g μ ν {\displaystyle g_{\mu \nu }} , since both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner. When fully written out, the EFE are a system of ten coupled, nonlinear, hyperbolic-elliptic partial differential equations . The above form of

3848-490: A university matriculation examination, and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated." General relativity can be understood by examining its similarities with and departures from classical physics. The first step

3996-539: A wave train traveling through empty space or Gowdy universes , varieties of an expanding cosmos filled with gravitational waves. But for gravitational waves produced in astrophysically relevant situations, such as the merger of two black holes, numerical methods are presently the only way to construct appropriate models. General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. It predicts an overall rotation ( precession ) of planetary orbits, as well as orbital decay caused by

4144-738: Is (+ − −) , Peebles (1980) and Efstathiou et al. (1990) are (− + +) , Rindler (1977), Atwater (1974), Collins Martin & Squires (1989) and Peacock (1999) are (− + −) . Authors including Einstein have used a different sign in their definition for the Ricci tensor which results in the sign of the constant on the right side being negative: R μ ν − 1 2 R g μ ν − Λ g μ ν = − κ T μ ν . {\displaystyle R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }-\Lambda g_{\mu \nu }=-\kappa T_{\mu \nu }.} The sign of

4292-526: Is Minkowskian , and the laws of physics exhibit local Lorentz invariance . The core concept of general-relativistic model-building is that of a solution of Einstein's equations . Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi- Riemannian manifold (usually defined by giving the metric in specific coordinates), and specific matter fields defined on that manifold. Matter and geometry must satisfy Einstein's equations, so in particular,

4440-693: Is a map which takes smooth vector fields X {\displaystyle X} , Y {\displaystyle Y} , and Z {\displaystyle Z} , and returns the vector field R ( X , Y ) Z := ∇ X ∇ Y Z − ∇ Y ∇ X Z − ∇ [ X , Y ] Z {\displaystyle R(X,Y)Z:=\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z-\nabla _{[X,Y]}Z} on vector fields X , Y , Z {\displaystyle X,Y,Z} . Since R {\displaystyle R}

4588-432: Is a natural by-product, would correspond to the full matrix of second derivatives of a function. However, there are other ways to draw the same analogy. In three-dimensional topology , the Ricci tensor contains all of the information which in higher dimensions is encoded by the more complicated Riemann curvature tensor . In part, this simplicity allows for the application of many geometric and analytic tools, which led to

Einstein field equations - Misplaced Pages Continue

4736-423: Is a scalar parameter of motion (e.g. the proper time ), and Γ μ α β {\displaystyle \Gamma ^{\mu }{}_{\alpha \beta }} are Christoffel symbols (sometimes called the affine connection coefficients or Levi-Civita connection coefficients) which is symmetric in the two lower indices. Greek indices may take the values: 0, 1, 2, 3 and

4884-397: Is a standard exercise of (multi)linear algebra to verify that this definition does not depend on the choice of the basis v 1 , … , v n {\displaystyle v_{1},\ldots ,v_{n}} . In abstract index notation , R i c a b = R c b c a = R c

5032-532: Is a tensor field, for each point p ∈ M {\displaystyle p\in M} , it gives rise to a (multilinear) map: R p : T p M × T p M × T p M → T p M . {\displaystyle \operatorname {R} _{p}:T_{p}M\times T_{p}M\times T_{p}M\to T_{p}M.} Define for each point p ∈ M {\displaystyle p\in M}

5180-445: Is a universality of free fall (also known as the weak equivalence principle , or the universal equality of inertial and passive-gravitational mass): the trajectory of a test body in free fall depends only on its position and initial speed, but not on any of its material properties. A simplified version of this is embodied in Einstein's elevator experiment , illustrated in the figure on the right: for an observer in an enclosed room, it

5328-476: Is achieved in approximating the spacetime as having only small deviations from flat spacetime , leading to the linearized EFE . These equations are used to study phenomena such as gravitational waves . The Einstein field equations (EFE) may be written in the form: where G μ ν {\displaystyle G_{\mu \nu }} is the Einstein tensor, g μ ν {\displaystyle g_{\mu \nu }}

5476-402: Is based on the propagation of light, and thus on electromagnetism, which could have a different set of preferred frames . But using different assumptions about the special-relativistic frames (such as their being earth-fixed, or in free fall), one can derive different predictions for the gravitational redshift, that is, the way in which the frequency of light shifts as the light propagates through

5624-499: Is curved. The resulting Newton–Cartan theory is a geometric formulation of Newtonian gravity using only covariant concepts, i.e. a description which is valid in any desired coordinate system. In this geometric description, tidal effects —the relative acceleration of bodies in free fall—are related to the derivative of the connection, showing how the modified geometry is caused by the presence of mass. As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics,

5772-650: Is defined as above. The existence of a cosmological constant is thus equivalent to the existence of a vacuum energy and a pressure of opposite sign. This has led to the terms "cosmological constant" and "vacuum energy" being used interchangeably in general relativity. General relativity is consistent with the local conservation of energy and momentum expressed as ∇ β T α β = T α β ; β = 0. {\displaystyle \nabla _{\beta }T^{\alpha \beta }={T^{\alpha \beta }}_{;\beta }=0.} Contracting

5920-405: Is defined in the absence of gravity. For practical applications, it is a suitable model whenever gravity can be neglected. Bringing gravity into play, and assuming the universality of free fall motion, an analogous reasoning as in the previous section applies: there are no global inertial frames . Instead there are approximate inertial frames moving alongside freely falling particles. Translated into

6068-408: Is determined by the sectional curvatures of a Riemannian manifold, but generally contains less information. Indeed, if ξ {\displaystyle \xi } is a vector of unit length on a Riemannian n {\displaystyle n} -manifold, then Ric ⁡ ( ξ , ξ ) {\displaystyle \operatorname {Ric} (\xi ,\xi )}

Einstein field equations - Misplaced Pages Continue

6216-411: Is equivalent to R β δ ; ε − R β ε ; δ + R γ β δ ε ; γ = 0 {\displaystyle R_{\beta \delta ;\varepsilon }-R_{\beta \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=0} using

6364-445: Is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is stationary in a gravitational field and the ball accelerating, or in free space aboard a rocket that is accelerating at a rate equal to that of the gravitational field versus the ball which upon release has nil acceleration. Given the universality of free fall, there is no observable distinction between inertial motion and motion under

6512-545: Is known as gravitational time dilation. Gravitational redshift has been measured in the laboratory and using astronomical observations. Gravitational time dilation in the Earth's gravitational field has been measured numerous times using atomic clocks , while ongoing validation is provided as a side effect of the operation of the Global Positioning System (GPS). Tests in stronger gravitational fields are provided by

6660-404: Is mass. In special relativity, mass turns out to be part of a more general quantity called the energy–momentum tensor , which includes both energy and momentum densities as well as stress : pressure and shear. Using the equivalence principle, this tensor is readily generalized to curved spacetime. Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that

6808-451: Is merely a limiting case of (special) relativistic mechanics. In the language of symmetry : where gravity can be neglected, physics is Lorentz invariant as in special relativity rather than Galilei invariant as in classical mechanics. (The defining symmetry of special relativity is the Poincaré group , which includes translations, rotations, boosts and reflections.) The differences between

6956-430: Is now associated with electrically charged black holes . In 1917, Einstein applied his theory to the universe as a whole, initiating the field of relativistic cosmology. In line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant —to match that observational presumption. By 1929, however, the work of Hubble and others had shown that

7104-456: Is often called the contracted second Bianchi identity. Near any point p {\displaystyle p} in a Riemannian manifold ( M , g ) {\displaystyle \left(M,g\right)} , one can define preferred local coordinates, called geodesic normal coordinates . These are adapted to the metric so that geodesics through p {\displaystyle p} correspond to straight lines through

7252-408: Is precisely ( n − 1 ) {\displaystyle (n-1)} times the average value of the sectional curvature, taken over all the 2-planes containing ξ {\displaystyle \xi } . There is an ( n − 2 ) {\displaystyle (n-2)} -dimensional family of such 2-planes, and so only in dimensions 2 and 3 does

7400-429: Is presently the subject of much research. Suppose that ( M , g ) {\displaystyle \left(M,g\right)} is an n {\displaystyle n} -dimensional Riemannian or pseudo-Riemannian manifold , equipped with its Levi-Civita connection ∇ {\displaystyle \nabla } . The Riemann curvature of M {\displaystyle M}

7548-465: Is related to the choice of convention for the Ricci tensor: R μ ν = [ S 2 ] × [ S 3 ] × R α μ α ν {\displaystyle R_{\mu \nu }=[S2]\times [S3]\times {R^{\alpha }}_{\mu \alpha \nu }} With these definitions Misner, Thorne, and Wheeler classify themselves as (+ + +) , whereas Weinberg (1972)

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7696-553: Is the Ricci curvature tensor , and R {\displaystyle R} is the scalar curvature . This is a symmetric second-degree tensor that depends on only the metric tensor and its first and second derivatives. The Einstein gravitational constant is defined as where G is the Newtonian constant of gravitation and c is the speed of light in vacuum . The EFE can thus also be written as In standard units, each term on

7844-487: Is the Shapiro Time Delay, the phenomenon that light signals take longer to move through a gravitational field than they would in the absence of that field. There have been numerous successful tests of this prediction. In the parameterized post-Newtonian formalism (PPN), measurements of both the deflection of light and the gravitational time delay determine a parameter called γ, which encodes the influence of gravity on

7992-477: Is the first derivative along i {\displaystyle i} th direction of R n {\displaystyle \mathbb {R} ^{n}} . This shows that the following definition does not depend on the choice of ( U , φ ) {\displaystyle \left(U,\varphi \right)} . For any p ∈ U {\displaystyle p\in U} , define

8140-897: Is the mass density. The orbit of a free-falling particle satisfies x → ¨ ( t ) = g → = − ∇ Φ ( x → ( t ) , t ) . {\displaystyle {\ddot {\vec {x}}}(t)={\vec {g}}=-\nabla \Phi \left({\vec {x}}(t),t\right)\,.} In tensor notation, these become Φ , i i = 4 π G ρ d 2 x i d t 2 = − Φ , i . {\displaystyle {\begin{aligned}\Phi _{,ii}&=4\pi G\rho \\{\frac {d^{2}x^{i}}{dt^{2}}}&=-\Phi _{,i}\,.\end{aligned}}} In general relativity, these equations are replaced by

8288-466: Is the metric tensor, T μ ν {\displaystyle T_{\mu \nu }} is the stress–energy tensor , Λ {\displaystyle \Lambda } is the cosmological constant and κ {\displaystyle \kappa } is the Einstein gravitational constant. The Einstein tensor is defined as where R μ ν {\displaystyle R_{\mu \nu }}

8436-409: Is the parametrized post-Newtonian (PPN) formalism, which allows quantitative comparisons between the predictions of general relativity and alternative theories. General relativity has a number of physical consequences. Some follow directly from the theory's axioms, whereas others have become clear only in the course of many years of research that followed Einstein's initial publication. Assuming that

8584-471: Is the realization that classical mechanics and Newton's law of gravity admit a geometric description. The combination of this description with the laws of special relativity results in a heuristic derivation of general relativity. At the base of classical mechanics is the notion that a body 's motion can be described as a combination of free (or inertial ) motion, and deviations from this free motion. Such deviations are caused by external forces acting on

8732-1069: Is the spacetime dimension. Solving for R and substituting this in the original EFE, one gets the following equivalent "trace-reversed" form: R μ ν − 2 D − 2 Λ g μ ν = κ ( T μ ν − 1 D − 2 T g μ ν ) . {\displaystyle R_{\mu \nu }-{\frac {2}{D-2}}\Lambda g_{\mu \nu }=\kappa \left(T_{\mu \nu }-{\frac {1}{D-2}}Tg_{\mu \nu }\right).} In D = 4 dimensions this reduces to R μ ν − Λ g μ ν = κ ( T μ ν − 1 2 T g μ ν ) . {\displaystyle R_{\mu \nu }-\Lambda g_{\mu \nu }=\kappa \left(T_{\mu \nu }-{\frac {1}{2}}T\,g_{\mu \nu }\right).} Reversing

8880-602: The curvature of spacetime is directly related to the energy and momentum of whatever present matter and radiation . The relation is specified by the Einstein field equations , a system of second-order partial differential equations . Newton's law of universal gravitation , which describes classical gravity, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond Newton's law of universal gravitation in classical physics . These predictions concern

9028-432: The Einstein notation , meaning that repeated indices are summed (i.e. from zero to three). The Christoffel symbols are functions of the four spacetime coordinates, and so are independent of the velocity or acceleration or other characteristics of a test particle whose motion is described by the geodesic equation. In general relativity, the effective gravitational potential energy of an object of mass m revolving around

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9176-600: The Gödel universe (which opens up the intriguing possibility of time travel in curved spacetimes), the Taub–NUT solution (a model universe that is homogeneous , but anisotropic ), and anti-de Sitter space (which has recently come to prominence in the context of what is called the Maldacena conjecture ). Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on

9324-543: The Taylor expansion of the metric applied to a Jacobi field along a radial geodesic in the normal coordinate system, one has g i j = δ i j − 1 3 R i k j l x k x l + O ( | x | 3 ) . {\displaystyle g_{ij}=\delta _{ij}-{\frac {1}{3}}R_{ikjl}x^{k}x^{l}+O\left(|x|^{3}\right).} In these coordinates,

9472-1511: The components of the tangent vectors at p {\displaystyle p} in X {\displaystyle X} and Y {\displaystyle Y} relative to the coordinate vector fields of ( U , φ ) {\displaystyle \left(U,\varphi \right)} . It is common to abbreviate the above formal presentation in the following style: Γ i j k := 1 2 g k l ( ∂ i g j l + ∂ j g i l − ∂ l g i j ) R j k := ∂ i Γ j k i − ∂ j Γ k i i + Γ i p i Γ j k p − Γ j p i Γ i k p . {\displaystyle {\begin{aligned}\Gamma _{ij}^{k}&:={\frac {1}{2}}g^{kl}\left(\partial _{i}g_{jl}+\partial _{j}g_{il}-\partial _{l}g_{ij}\right)\\R_{jk}&:=\partial _{i}\Gamma _{jk}^{i}-\partial _{j}\Gamma _{ki}^{i}+\Gamma _{ip}^{i}\Gamma _{jk}^{p}-\Gamma _{jp}^{i}\Gamma _{ik}^{p}.\end{aligned}}} It can be directly checked that R j k = R ~

9620-843: The differential Bianchi identity R α β [ γ δ ; ε ] = 0 {\displaystyle R_{\alpha \beta [\gamma \delta ;\varepsilon ]}=0} with g gives, using the fact that the metric tensor is covariantly constant, i.e. g ;γ = 0 , R γ β γ δ ; ε + R γ β ε γ ; δ + R γ β δ ε ; γ = 0 {\displaystyle {R^{\gamma }}_{\beta \gamma \delta ;\varepsilon }+{R^{\gamma }}_{\beta \varepsilon \gamma ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=0} The antisymmetry of

9768-631: The field equation for gravity relates this tensor and the Ricci tensor , which describes a particular class of tidal effects: the change in volume for a small cloud of test particles that are initially at rest, and then fall freely. In special relativity, conservation of energy –momentum corresponds to the statement that the energy–momentum tensor is divergence -free. This formula, too, is readily generalized to curved spacetime by replacing partial derivatives with their curved- manifold counterparts, covariant derivatives studied in differential geometry. With this additional condition—the covariant divergence of

9916-472: The general theory of relativity , and as Einstein's theory of gravity , is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics . General relativity generalizes special relativity and refines Newton's law of universal gravitation , providing a unified description of gravity as a geometric property of space and time , or four-dimensional spacetime . In particular,

10064-473: The post-Newtonian expansion , both of which were developed by Einstein. The latter provides a systematic approach to solving for the geometry of a spacetime that contains a distribution of matter that moves slowly compared with the speed of light. The expansion involves a series of terms; the first terms represent Newtonian gravity, whereas the later terms represent ever smaller corrections to Newton's theory due to general relativity. An extension of this expansion

10212-454: The scalar gravitational potential of classical physics by a symmetric rank -two tensor , the latter reduces to the former in certain limiting cases . For weak gravitational fields and slow speed relative to the speed of light, the theory's predictions converge on those of Newton's law of universal gravitation. As it is constructed using tensors, general relativity exhibits general covariance : its laws—and further laws formulated within

10360-490: The solution of the Poincaré conjecture through the work of Richard S. Hamilton and Grigori Perelman . In differential geometry, lower bounds on the Ricci tensor on a Riemannian manifold allow one to extract global geometric and topological information by comparison (cf. comparison theorem ) with the geometry of a constant curvature space form . This is since lower bounds on the Ricci tensor can be successfully used in studying

10508-417: The stress–energy tensor ). Analogously to the way that electromagnetic fields are related to the distribution of charges and currents via Maxwell's equations , the EFE relate the spacetime geometry to the distribution of mass–energy, momentum and stress, that is, they determine the metric tensor of spacetime for a given arrangement of stress–energy–momentum in the spacetime. The relationship between

10656-429: The summation convention is used for repeated indices α {\displaystyle \alpha } and β {\displaystyle \beta } . The quantity on the left-hand-side of this equation is the acceleration of a particle, and so this equation is analogous to Newton's laws of motion which likewise provide formulae for the acceleration of a particle. This equation of motion employs

10804-516: The Cheng-Yau and Li-Yau inequalities) nearly always depend on a lower bound for the Ricci curvature. In 2007, John Lott , Karl-Theodor Sturm , and Cedric Villani demonstrated decisively that lower bounds on Ricci curvature can be understood entirely in terms of the metric space structure of a Riemannian manifold, together with its volume form. This established a deep link between Ricci curvature and Wasserstein geometry and optimal transport , which

10952-1613: The EFE is the standard established by Misner, Thorne, and Wheeler (MTW). The authors analyzed conventions that exist and classified these according to three signs ([S1] [S2] [S3]): g μ ν = [ S 1 ] × diag ⁡ ( − 1 , + 1 , + 1 , + 1 ) R μ α β γ = [ S 2 ] × ( Γ α γ , β μ − Γ α β , γ μ + Γ σ β μ Γ γ α σ − Γ σ γ μ Γ β α σ ) G μ ν = [ S 3 ] × κ T μ ν {\displaystyle {\begin{aligned}g_{\mu \nu }&=[S1]\times \operatorname {diag} (-1,+1,+1,+1)\\[6pt]{R^{\mu }}_{\alpha \beta \gamma }&=[S2]\times \left(\Gamma _{\alpha \gamma ,\beta }^{\mu }-\Gamma _{\alpha \beta ,\gamma }^{\mu }+\Gamma _{\sigma \beta }^{\mu }\Gamma _{\gamma \alpha }^{\sigma }-\Gamma _{\sigma \gamma }^{\mu }\Gamma _{\beta \alpha }^{\sigma }\right)\\[6pt]G_{\mu \nu }&=[S3]\times \kappa T_{\mu \nu }\end{aligned}}} The third sign above

11100-440: The EFE reduce to Newton's law of gravitation in the limit of a weak gravitational field and velocities that are much less than the speed of light . Exact solutions for the EFE can only be found under simplifying assumptions such as symmetry . Special classes of exact solutions are most often studied since they model many gravitational phenomena, such as rotating black holes and the expanding universe . Further simplification

11248-896: The Einstein field equations in the trace-reversed form R μ ν = K ( T μ ν − 1 2 T g μ ν ) {\displaystyle R_{\mu \nu }=K\left(T_{\mu \nu }-{\tfrac {1}{2}}Tg_{\mu \nu }\right)} for some constant, K , and the geodesic equation d 2 x α d τ 2 = − Γ β γ α d x β d τ d x γ d τ . {\displaystyle {\frac {d^{2}x^{\alpha }}{d\tau ^{2}}}=-\Gamma _{\beta \gamma }^{\alpha }{\frac {dx^{\beta }}{d\tau }}{\frac {dx^{\gamma }}{d\tau }}\,.} To see how

11396-439: The Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in n dimensions. The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when T μν is everywhere zero) define Einstein manifolds . The equations are more complex than they appear. Given

11544-483: The Newtonian limit and treating the orbiting body as a test particle . For him, the fact that his theory gave a straightforward explanation of Mercury's anomalous perihelion shift, discovered earlier by Urbain Le Verrier in 1859, was important evidence that he had at last identified the correct form of the gravitational field equations. Ricci curvature The Ricci tensor can be characterized by measurement of how

11692-986: The Ricci curvature tensor and the scalar curvature then show that R ; ε − 2 R γ ε ; γ = 0 {\displaystyle R_{;\varepsilon }-2{R^{\gamma }}_{\varepsilon ;\gamma }=0} which can be rewritten as ( R γ ε − 1 2 g γ ε R ) ; γ = 0 {\displaystyle \left({R^{\gamma }}_{\varepsilon }-{\tfrac {1}{2}}{g^{\gamma }}_{\varepsilon }R\right)_{;\gamma }=0} A final contraction with g gives ( R γ δ − 1 2 g γ δ R ) ; γ = 0 {\displaystyle \left(R^{\gamma \delta }-{\tfrac {1}{2}}g^{\gamma \delta }R\right)_{;\gamma }=0} which by

11840-471: The Ricci tensor determine the full curvature tensor. A notable exception is when the manifold is given a priori as a hypersurface of Euclidean space . The second fundamental form , which determines the full curvature via the Gauss–Codazzi equation , is itself determined by the Ricci tensor and the principal directions of the hypersurface are also the eigendirections of the Ricci tensor. The tensor

11988-441: The Ricci tensor is completely determined by knowing the quantity Ric ⁡ ( X , X ) {\displaystyle \operatorname {Ric} (X,X)} for all vectors X {\displaystyle X} of unit length. This function on the set of unit tangent vectors is often also called the Ricci curvature , since knowing it is equivalent to knowing the Ricci curvature tensor. The Ricci curvature

12136-644: The Riemann tensor allows the second term in the above expression to be rewritten: R γ β γ δ ; ε − R γ β γ ε ; δ + R γ β δ ε ; γ = 0 {\displaystyle {R^{\gamma }}_{\beta \gamma \delta ;\varepsilon }-{R^{\gamma }}_{\beta \gamma \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=0} which

12284-1800: The Riemann tensor, they do not differ about the Ricci tensor. Let ( M , g ) {\displaystyle \left(M,g\right)} be a smooth Riemannian or pseudo-Riemannian n {\displaystyle n} -manifold. Given a smooth chart ( U , φ ) {\displaystyle \left(U,\varphi \right)} one then has functions g i j : φ ( U ) → R {\displaystyle g_{ij}:\varphi (U)\rightarrow \mathbb {R} } and g i j : φ ( U ) → R {\displaystyle g^{ij}:\varphi (U)\rightarrow \mathbb {R} } for each i , j = 1 , … , n {\displaystyle i,j=1,\ldots ,n} which satisfy ∑ k = 1 n g i k ( x ) g k j ( x ) = δ j i = { 1 i = j 0 i ≠ j {\displaystyle \sum _{k=1}^{n}g^{ik}(x)g_{kj}(x)=\delta _{j}^{i}={\begin{cases}1&i=j\\0&i\neq j\end{cases}}} for all x ∈ φ ( U ) {\displaystyle x\in \varphi (U)} . The latter shows that, expressed as matrices, g i j ( x ) = ( g − 1 ) i j ( x ) {\displaystyle g^{ij}(x)=(g^{-1})_{ij}(x)} . The functions g i j {\displaystyle g_{ij}} are defined by evaluating g {\displaystyle g} on coordinate vector fields, while

12432-413: The actual motions of bodies and making allowances for the external forces (such as electromagnetism or friction ), can be used to define the geometry of space, as well as a time coordinate . However, there is an ambiguity once gravity comes into play. According to Newton's law of gravity, and independently verified by experiments such as that of Eötvös and its successors (see Eötvös experiment ), there

12580-401: The base of cosmological models of an expanding universe . Widely acknowledged as a theory of extraordinary beauty , general relativity has often been described as the most beautiful of all existing physical theories. Henri Poincaré 's 1905 theory of the dynamics of the electron was a relativistic theory which he applied to all forces, including gravity. While others thought that gravity

12728-406: The connection that satisfies the equivalence principle and makes space locally Minkowskian (that is, in suitable locally inertial coordinates , the metric is Minkowskian, and its first partial derivatives and the connection coefficients vanish). Having formulated the relativistic, geometric version of the effects of gravity, the question of gravity's source remains. In Newtonian gravity, the source

12876-423: The core of the mathematical formulation of general relativity . The EFE is a tensor equation relating a set of symmetric 4 × 4 tensors . Each tensor has 10 independent components. The four Bianchi identities reduce the number of independent equations from 10 to 6, leaving the metric with four gauge-fixing degrees of freedom , which correspond to the freedom to choose a coordinate system. Although

13024-403: The cosmological constant was almost universally assumed to be zero. More recent astronomical observations have shown an accelerating expansion of the universe , and to explain this a positive value of Λ is needed. The effect of the cosmological constant is negligible at the scale of a galaxy or smaller. Einstein thought of the cosmological constant as an independent parameter, but its term in

13172-483: The cosmological term would change in both these versions if the (+ − − −) metric sign convention is used rather than the MTW (− + + +) metric sign convention adopted here. Taking the trace with respect to the metric of both sides of the EFE one gets R − D 2 R + D Λ = κ T , {\displaystyle R-{\frac {D}{2}}R+D\Lambda =\kappa T,} where D

13320-1006: The definition of the Ricci tensor . Next, contract again with the metric g β δ ( R β δ ; ε − R β ε ; δ + R γ β δ ε ; γ ) = 0 {\displaystyle g^{\beta \delta }\left(R_{\beta \delta ;\varepsilon }-R_{\beta \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }\right)=0} to get R δ δ ; ε − R δ ε ; δ + R γ δ δ ε ; γ = 0 {\displaystyle {R^{\delta }}_{\delta ;\varepsilon }-{R^{\delta }}_{\varepsilon ;\delta }+{R^{\gamma \delta }}_{\delta \varepsilon ;\gamma }=0} The definitions of

13468-585: The definitions directly using local coordinates are preferable, since the "crucial property" of the Riemann tensor mentioned above requires M {\displaystyle M} to be Hausdorff in order to hold. By contrast, the local coordinate approach only requires a smooth atlas. It is also somewhat easier to connect the "invariance" philosophy underlying the local approach with the methods of constructing more exotic geometric objects, such as spinor fields . The complicated formula defining R i j {\displaystyle R_{ij}} in

13616-557: The deflection of starlight by the Sun during the total solar eclipse of 29 May 1919 , instantly making Einstein famous. Yet the theory remained outside the mainstream of theoretical physics and astrophysics until developments between approximately 1960 and 1975, now known as the golden age of general relativity . Physicists began to understand the concept of a black hole, and to identify quasars as one of these objects' astrophysical manifestations. Ever more precise solar system tests confirmed

13764-446: The distortions along the principal axes counteract one another. The Ricci curvature would then vanish along ξ {\displaystyle \xi } . In physical applications, the presence of a nonvanishing sectional curvature does not necessarily indicate the presence of any mass locally; if an initially circular cross-section of a cone of worldlines later becomes elliptical, without changing its volume, then this

13912-452: The emission of gravitational waves and effects related to the relativity of direction. In general relativity, the apsides of any orbit (the point of the orbiting body's closest approach to the system's center of mass ) will precess ; the orbit is not an ellipse , but akin to an ellipse that rotates on its focus, resulting in a rose curve -like shape (see image). Einstein first derived this result by using an approximate metric representing

14060-500: The end-state for massive stars . Microquasars and active galactic nuclei are believed to be stellar black holes and supermassive black holes . It also predicts gravitational lensing , where the bending of light results in multiple images of the same distant astronomical phenomenon. Other predictions include the existence of gravitational waves , which have been observed directly by the physics collaboration LIGO and other observatories. In addition, general relativity has provided

14208-555: The energy–momentum tensor, and hence of whatever is on the other side of the equation, is zero—the simplest nontrivial set of equations are what are called Einstein's (field) equations: G μ ν ≡ R μ ν − 1 2 R g μ ν = κ T μ ν {\displaystyle G_{\mu \nu }\equiv R_{\mu \nu }-{\textstyle 1 \over 2}R\,g_{\mu \nu }=\kappa T_{\mu \nu }\,} On

14356-446: The equivalence principle holds, gravity influences the passage of time. Light sent down into a gravity well is blueshifted , whereas light sent in the opposite direction (i.e., climbing out of the gravity well) is redshifted ; collectively, these two effects are known as the gravitational frequency shift. More generally, processes close to a massive body run more slowly when compared with processes taking place farther away; this effect

14504-456: The exceedingly weak waves that are expected to arrive here on Earth from far-off cosmic events, which typically result in relative distances increasing and decreasing by 10 − 21 {\displaystyle 10^{-21}} or less. Data analysis methods routinely make use of the fact that these linearized waves can be Fourier decomposed . Some exact solutions describe gravitational waves without any approximation, e.g.,

14652-408: The exterior Schwarzschild solution or, for more than a single mass, the post-Newtonian expansion), several effects of gravity on light propagation emerge. Although the bending of light can also be derived by extending the universality of free fall to light, the angle of deflection resulting from such calculations is only half the value given by general relativity. Closely related to light deflection

14800-430: The field equation can also be moved algebraically to the other side and incorporated as part of the stress–energy tensor: T μ ν ( v a c ) = − Λ κ g μ ν . {\displaystyle T_{\mu \nu }^{\mathrm {(vac)} }=-{\frac {\Lambda }{\kappa }}g_{\mu \nu }\,.} This tensor describes

14948-433: The first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric . This solution laid the groundwork for the description of the final stages of gravitational collapse, and the objects known today as black holes. In the same year, the first steps towards generalizing Schwarzschild's solution to electrically charged objects were taken, eventually resulting in the Reissner–Nordström solution , which

15096-515: The functions Γ a b c := 1 2 ∑ d = 1 n ( ∂ g b d ∂ x a + ∂ g a d ∂ x b − ∂ g a b ∂ x d ) g c d R i j := ∑

15244-625: The functions g i j {\displaystyle g^{ij}} are defined so that, as a matrix-valued function, they provide an inverse to the matrix-valued function x ↦ g i j ( x ) {\displaystyle x\mapsto g_{ij}(x)} . Now define, for each a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} , i {\displaystyle i} , and j {\displaystyle j} between 1 and n {\displaystyle n} ,

15392-487: The functions computed as above via the chart ( U , φ ) {\displaystyle \left(U,\varphi \right)} and let r i j : ψ ( V ) → R {\displaystyle r_{ij}:\psi (V)\rightarrow \mathbb {R} } be the functions computed as above via the chart ( V , ψ ) {\displaystyle \left(V,\psi \right)} . Then one can check by

15540-412: The general relativistic framework—take on the same form in all coordinate systems . Furthermore, the theory does not contain any invariant geometric background structures, i.e. it is background independent . It thus satisfies a more stringent general principle of relativity , namely that the laws of physics are the same for all observers. Locally , as expressed in the equivalence principle, spacetime

15688-477: The geometry of space. Predicted in 1916 by Albert Einstein, there are gravitational waves: ripples in the metric of spacetime that propagate at the speed of light. These are one of several analogies between weak-field gravity and electromagnetism in that, they are analogous to electromagnetic waves . On 11 February 2016, the Advanced LIGO team announced that they had directly detected gravitational waves from

15836-464: The history of the universe and have provided the modern framework for cosmology , thus leading to the discovery of the Big Bang and cosmic microwave background radiation. Despite the introduction of a number of alternative theories , general relativity continues to be the simplest theory consistent with experimental data . Reconciliation of general relativity with the laws of quantum physics remains

15984-441: The image), and a set of events for which such an influence is impossible (such as event C in the image). These sets are observer -independent. In conjunction with the world-lines of freely falling particles, the light-cones can be used to reconstruct the spacetime's semi-Riemannian metric, at least up to a positive scalar factor. In mathematical terms, this defines a conformal structure or conformal geometry. Special relativity

16132-446: The influence of the gravitational force. This suggests the definition of a new class of inertial motion, namely that of objects in free fall under the influence of gravity. This new class of preferred motions, too, defines a geometry of space and time—in mathematical terms, it is the geodesic motion associated with a specific connection which depends on the gradient of the gravitational potential . Space, in this construction, still has

16280-481: The informal notation. The two above definitions are identical. The formulas defining Γ i j k {\displaystyle \Gamma _{ij}^{k}} and R i j {\displaystyle R_{ij}} in the coordinate approach have an exact parallel in the formulas defining the Levi-Civita connection, and the Riemann curvature via the Levi-Civita connection. Arguably,

16428-813: The introductory section is the same as that in the following section. The only difference is that terms have been grouped so that it is easy to see that R i j = R j i . {\displaystyle R_{ij}=R_{ji}.} As can be seen from the symmetries of the Riemann curvature tensor, the Ricci tensor of a Riemannian manifold is symmetric , in the sense that Ric ⁡ ( X , Y ) = Ric ⁡ ( Y , X ) {\displaystyle \operatorname {Ric} (X,Y)=\operatorname {Ric} (Y,X)} for all X , Y ∈ T p M . {\displaystyle X,Y\in T_{p}M.} It thus follows linear-algebraically that

16576-417: The key mathematical framework on which he fit his physical ideas of gravity. This idea was pointed out by mathematician Marcel Grossmann and published by Grossmann and Einstein in 1913. The Einstein field equations are nonlinear and considered difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But in 1916, the astrophysicist Karl Schwarzschild found

16724-410: The language of spacetime: the straight time-like lines that define a gravity-free inertial frame are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity necessitates a change in spacetime geometry. A priori, it is not clear whether the new local frames in free fall coincide with the reference frames in which the laws of special relativity hold—that theory

16872-601: The latter reduces to the former, we assume that the test particle's velocity is approximately zero d x β d τ ≈ ( d t d τ , 0 , 0 , 0 ) {\displaystyle {\frac {dx^{\beta }}{d\tau }}\approx \left({\frac {dt}{d\tau }},0,0,0\right)} and thus d d t ( d t d τ ) ≈ 0 {\displaystyle {\frac {d}{dt}}\left({\frac {dt}{d\tau }}\right)\approx 0} and that

17020-474: The left has units of 1/length. The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the stress–energy–momentum content of spacetime. The EFE can then be interpreted as a set of equations dictating how stress–energy–momentum determines the curvature of spacetime. These equations, together with the geodesic equation , which dictates how freely falling matter moves through spacetime, form

17168-405: The left-hand side is the Einstein tensor , G μ ν {\displaystyle G_{\mu \nu }} , which is symmetric and a specific divergence-free combination of the Ricci tensor R μ ν {\displaystyle R_{\mu \nu }} and the metric. In particular, is the curvature scalar. The Ricci tensor itself is related to

17316-562: The length functional in Riemannian geometry, as first shown in 1941 via Myers's theorem . One common source of the Ricci tensor is that it arises whenever one commutes the covariant derivative with the tensor Laplacian. This, for instance, explains its presence in the Bochner formula , which is used ubiquitously in Riemannian geometry. For example, this formula explains why the gradient estimates due to Shing-Tung Yau (and their developments such as

17464-474: The light of stars or distant quasars being deflected as it passes the Sun . This and related predictions follow from the fact that light follows what is called a light-like or null geodesic —a generalization of the straight lines along which light travels in classical physics. Such geodesics are the generalization of the invariance of lightspeed in special relativity. As one examines suitable model spacetimes (either

17612-465: The local conservation of stress–energy. This conservation law is a physical requirement. With his field equations Einstein ensured that general relativity is consistent with this conservation condition. The nonlinearity of the EFE distinguishes general relativity from many other fundamental physical theories. For example, Maxwell's equations of electromagnetism are linear in the electric and magnetic fields , and charge and current distributions (i.e.

17760-829: The map Ric p : T p M × T p M → R {\displaystyle \operatorname {Ric} _{p}:T_{p}M\times T_{p}M\to \mathbb {R} } by Ric p ⁡ ( Y , Z ) := tr ⁡ ( X ↦ R p ⁡ ( X , Y ) Z ) . {\displaystyle \operatorname {Ric} _{p}(Y,Z):=\operatorname {tr} {\big (}X\mapsto \operatorname {R} _{p}(X,Y)Z{\big )}.} That is, having fixed Y {\displaystyle Y} and Z {\displaystyle Z} , then for any orthonormal basis v 1 , … , v n {\displaystyle v_{1},\ldots ,v_{n}} of

17908-502: The matter content of the universe. Like the metric tensor, the Ricci tensor assigns to each tangent space of the manifold a symmetric bilinear form ( Besse 1987 , p. 43). Broadly, one could analogize the role of the Ricci curvature in Riemannian geometry to that of the Laplacian in the analysis of functions; in this analogy, the Riemann curvature tensor , of which the Ricci curvature

18056-455: The matter's energy–momentum tensor must be divergence-free. The matter must, of course, also satisfy whatever additional equations were imposed on its properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present. Einstein's equations are nonlinear partial differential equations and, as such, difficult to solve exactly. Nevertheless,

18204-498: The metric volume element then has the following expansion at p : d μ g = [ 1 − 1 6 R j k x j x k + O ( | x | 3 ) ] d μ Euclidean , {\displaystyle d\mu _{g}=\left[1-{\frac {1}{6}}R_{jk}x^{j}x^{k}+O\left(|x|^{3}\right)\right]d\mu _{\text{Euclidean}},} which follows by expanding

18352-1119: The metric and its derivatives are approximately static and that the squares of deviations from the Minkowski metric are negligible. Applying these simplifying assumptions to the spatial components of the geodesic equation gives d 2 x i d t 2 ≈ − Γ 00 i {\displaystyle {\frac {d^{2}x^{i}}{dt^{2}}}\approx -\Gamma _{00}^{i}} where two factors of ⁠ dt / dτ ⁠ have been divided out. This will reduce to its Newtonian counterpart, provided Φ , i ≈ Γ 00 i = 1 2 g i α ( g α 0 , 0 + g 0 α , 0 − g 00 , α ) . {\displaystyle \Phi _{,i}\approx \Gamma _{00}^{i}={\tfrac {1}{2}}g^{i\alpha }\left(g_{\alpha 0,0}+g_{0\alpha ,0}-g_{00,\alpha }\right)\,.} General relativity General relativity , also known as

18500-418: The metric tensor and the Einstein tensor allows the EFE to be written as a set of nonlinear partial differential equations when used in this way. The solutions of the EFE are the components of the metric tensor. The inertial trajectories of particles and radiation ( geodesics ) in the resulting geometry are then calculated using the geodesic equation . As well as implying local energy–momentum conservation,

18648-442: The more general Riemann curvature tensor as On the right-hand side, κ {\displaystyle \kappa } is a constant and T μ ν {\displaystyle T_{\mu \nu }} is the energy–momentum tensor. All tensors are written in abstract index notation . Matching the theory's prediction to observational results for planetary orbits or, equivalently, assuring that

18796-432: The observation of binary pulsars . All results are in agreement with general relativity. However, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid. General relativity predicts that the path of light will follow the curvature of spacetime as it passes near a star. This effect was initially confirmed by observing

18944-459: The ordinary Euclidean geometry . However, space time as a whole is more complicated. As can be shown using simple thought experiments following the free-fall trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (time-like vectors) will vary with the particle's trajectory; mathematically speaking, the Newtonian connection is not integrable . From this, one can deduce that spacetime

19092-561: The origin, in such a manner that the geodesic distance from p {\displaystyle p} corresponds to the Euclidean distance from the origin. In these coordinates, the metric tensor is well-approximated by the Euclidean metric, in the precise sense that g i j = δ i j + O ( | x | 2 ) . {\displaystyle g_{ij}=\delta _{ij}+O\left(|x|^{2}\right).} In fact, by taking

19240-497: The passage of time, the geometry of space, the motion of bodies in free fall , and the propagation of light, and include gravitational time dilation , gravitational lensing , the gravitational redshift of light, the Shapiro time delay and singularities / black holes . So far, all tests of general relativity have been shown to be in agreement with the theory. The time-dependent solutions of general relativity enable us to talk about

19388-504: The preface to Relativity: The Special and the General Theory , Einstein said "The present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. The work presumes a standard of education corresponding to that of

19536-430: The principle of equivalence and his sense that a proper description of gravity should be geometrical at its basis, so that there was an "element of revelation" in the manner in which Einstein arrived at his theory. Other elements of beauty associated with the general theory of relativity are its simplicity and symmetry, the manner in which it incorporates invariance and unification, and its perfect logical consistency. In

19684-446: The same premises, which include additional rules and/or constraints, leading to different field equations. Examples are Whitehead's theory , Brans–Dicke theory , teleparallelism , f ( R ) gravity and Einstein–Cartan theory . The derivation outlined in the previous section contains all the information needed to define general relativity, describe its key properties, and address a question of crucial importance in physics, namely how

19832-472: The speed of light in vacuum. When there is no matter present, so that the energy–momentum tensor vanishes, the results are the vacuum Einstein equations, In general relativity, the world line of a particle free from all external, non-gravitational force is a particular type of geodesic in curved spacetime. In other words, a freely moving or falling particle always moves along a geodesic. The geodesic equation is: where s {\displaystyle s}

19980-607: The square root of the determinant of the metric. Thus, if the Ricci curvature Ric ⁡ ( ξ , ξ ) {\displaystyle \operatorname {Ric} (\xi ,\xi )} is positive in the direction of a vector ξ {\displaystyle \xi } , the conical region in M {\displaystyle M} swept out by a tightly focused family of geodesic segments of length ε {\displaystyle \varepsilon } emanating from p {\displaystyle p} , with initial velocity inside

20128-419: The sum of two solutions is also a solution); another example is Schrödinger's equation of quantum mechanics , which is linear in the wavefunction . The EFE reduce to Newton's law of gravity by using both the weak-field approximation and the slow-motion approximation . In fact, the constant G appearing in the EFE is determined by making these two approximations. Newtonian gravitation can be written as

20276-586: The symmetry of the bracketed term and the definition of the Einstein tensor , gives, after relabelling the indices, G α β ; β = 0 {\displaystyle {G^{\alpha \beta }}_{;\beta }=0} Using the EFE, this immediately gives, ∇ β T α β = T α β ; β = 0 {\displaystyle \nabla _{\beta }T^{\alpha \beta }={T^{\alpha \beta }}_{;\beta }=0} which expresses

20424-504: The term containing the cosmological constant Λ was absent from the version in which he originally published them. Einstein then included the term with the cosmological constant to allow for a universe that is not expanding or contracting . This effort was unsuccessful because: Einstein then abandoned Λ , remarking to George Gamow "that the introduction of the cosmological term was the biggest blunder of his life". The inclusion of this term does not create inconsistencies. For many years

20572-518: The theory can be used for model-building. General relativity is a metric theory of gravitation. At its core are Einstein's equations , which describe the relation between the geometry of a four-dimensional pseudo-Riemannian manifold representing spacetime, and the energy–momentum contained in that spacetime. Phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as free-fall , orbital motion, and spacecraft trajectories ), correspond to inertial motion within

20720-486: The theory of a scalar field, Φ , which is the gravitational potential in joules per kilogram of the gravitational field g = −∇Φ , see Gauss's law for gravity ∇ 2 Φ ( x → , t ) = 4 π G ρ ( x → , t ) {\displaystyle \nabla ^{2}\Phi \left({\vec {x}},t\right)=4\pi G\rho \left({\vec {x}},t\right)} where ρ

20868-644: The theory's predictive power, and relativistic cosmology also became amenable to direct observational tests. General relativity has acquired a reputation as a theory of extraordinary beauty. Subrahmanyan Chandrasekhar has noted that at multiple levels, general relativity exhibits what Francis Bacon has termed a "strangeness in the proportion" ( i.e . elements that excite wonderment and surprise). It juxtaposes fundamental concepts (space and time versus matter and motion) which had previously been considered as entirely independent. Chandrasekhar also noted that Einstein's only guides in his search for an exact theory were

21016-651: The trace again would restore the original EFE. The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace g μ ν {\displaystyle g_{\mu \nu }} in the expression on the right with the Minkowski metric without significant loss of accuracy). In the Einstein field equations G μ ν + Λ g μ ν = κ T μ ν , {\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }\,,}

21164-487: The two become significant when dealing with speeds approaching the speed of light , and with high-energy phenomena. With Lorentz symmetry, additional structures come into play. They are defined by the set of light cones (see image). The light-cones define a causal structure: for each event A , there is a set of events that can, in principle, either influence or be influenced by A via signals or interactions that do not need to travel faster than light (such as event B in

21312-489: The universe is expanding. This is readily described by the expanding cosmological solutions found by Friedmann in 1922, which do not require a cosmological constant. Lemaître used these solutions to formulate the earliest version of the Big Bang models, in which the universe has evolved from an extremely hot and dense earlier state. Einstein later declared the cosmological constant the biggest blunder of his life. During that period, general relativity remained something of

21460-464: The vector space T p M {\displaystyle T_{p}M} , one has Ric p ⁡ ( Y , Z ) = ∑ i = 1 ⟨ R p ⁡ ( v i , Y ) Z , v i ⟩ . {\displaystyle \operatorname {Ric} _{p}(Y,Z)=\sum _{i=1}\langle \operatorname {R} _{p}(v_{i},Y)Z,v_{i}\rangle .} It

21608-438: The weak-gravity, low-speed limit is Newtonian mechanics, the proportionality constant κ {\displaystyle \kappa } is found to be κ = 8 π G c 4 {\textstyle \kappa ={\frac {8\pi G}{c^{4}}}} , where G {\displaystyle G} is the Newtonian constant of gravitation and c {\displaystyle c}

21756-474: Was instantaneous or of electromagnetic origin, he suggested that relativity was "something due to our methods of measurement". In his theory, he showed that gravitational waves propagate at the speed of light. Soon afterwards, Einstein started thinking about how to incorporate gravity into his relativistic framework. In 1907, beginning with a simple thought experiment involving an observer in free fall (FFO), he embarked on what would be an eight-year search for

21904-486: Was introduced by Ricci for this reason. As can be seen from the second Bianchi identity, one has div ⁡ Ric = 1 2 d R , {\displaystyle \operatorname {div} \operatorname {Ric} ={\frac {1}{2}}dR,} where R {\displaystyle R} is the scalar curvature , defined in local coordinates as g i j R i j . {\displaystyle g^{ij}R_{ij}.} This

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